Quartic residues and binary quadratic forms

Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let ε_d be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that ε_d is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {u_n(a,b)} is the Lucas sequence defined by and .     

When subset-sums do not cover all the residues modulo p

Let . We prove that a subset of , where p is a prime number, with cardinality larger than such that its subset sums do not cover has an automorphic image which is rather concentrated; more precisely, there exists s prime to p such that

     

Volumes and diophantine inequalities associated with decomposable forms

For homogeneous decomposable forms in n variables with real coefficients, we consider the associated volume of all real solutions to the inequality . We relate this to the number of integral solutions to the Diophantine inequality in the case where F has rational coefficients. We find quantities which bound the volume and which yield good upper bounds on the number of solutions to the Diophantine inequality in the rational case.     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

Enumeration of isomorphism classes of extensions of p-adic fields

Let Ω be an algebraic closure of and let F be a finite extension of contained in Ω. Given positive integers f and e, the number of extensions K/F contained in Ω with residue degree f and ramification index e was computed by Krasner. This paper is concerned with the number of F-isomorphism classes of such extensions. We determine completely when p2?e and get partial results when p2||e. When s is large, is equal to the number of isomorphism classes of finite commutative chain rings with residue field , ramification index e, and length s.     

The distribution of patterns of inverses modulo a prime

We investigate the distribution of the numbers x∈[1,p] for which all lie in a subset of the set of multiplicative inverses modulo a prime p. Here the a_i are integers coprime to p and the numbers are distinct .     

Realizable classes of tetrahedral extensions

Let k be a number field and O_k its ring of integers. Let Γ be the alternating group A₄. Let be a maximal O_k-order in k[Γ] containing O_k[Γ] and its class group. We denote by the set of realizable classes, that is the set of classes such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Γ, for which the class of is equal to c, where O_N is the ring of integers of N. In this article we determine and we prove that it is a subgroup of provided that k and the 3rd cyclotomic field of are linearly disjoint, and the class number of k is odd.